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Close connectedness of the moduli stack of reduced curves

Sebastian Bozlee

Abstract

We prove that the moduli stack of all reduced $n$-pointed curves is ``closely connected" in characteristic zero, in the sense that each irreducible component of the stack intersects the component of smoothable curves. We achieve this by performing a detailed study of Ishii's territories, moduli schemes parametrizing reduced curve singularities together with a normalization map. We give explicit equations for territories, bound their dimensions, describe certain functoriality properties, and study the action of several groups on territories. Along the way, we prove the existence of nonsmoothable reduced curve singularities in new ranges, generalizing work of Mumford, Pinkham, Greuel, and Stevens.

Close connectedness of the moduli stack of reduced curves

Abstract

We prove that the moduli stack of all reduced -pointed curves is ``closely connected" in characteristic zero, in the sense that each irreducible component of the stack intersects the component of smoothable curves. We achieve this by performing a detailed study of Ishii's territories, moduli schemes parametrizing reduced curve singularities together with a normalization map. We give explicit equations for territories, bound their dimensions, describe certain functoriality properties, and study the action of several groups on territories. Along the way, we prove the existence of nonsmoothable reduced curve singularities in new ranges, generalizing work of Mumford, Pinkham, Greuel, and Stevens.

Paper Structure

This paper contains 14 sections, 37 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Basic theory
  4. Definition and basic properties of territories
  5. Equations for territories
  6. Based territories
  7. The spine
  8. Construction and dimension bounds
  9. The union of spines
  10. Application to smoothability of singularities
  11. Restriction, contraction, and join of branches
  12. Gluing restrictions of singularities
  13. The torus action, limits, and vanishing sequences
  14. The action of $\mathrm{Aut}(A_{\mathbf{c}}^+)$ and close connectedness of $\mathcal{U}_{g,n}$

Key Result

Theorem 1.1

Let $k$ be a field of characteristic 0. Then each irreducible component of $\mathcal{U}_{g,n} \times \mathrm{Spec}\, k$ intersects the irreducible substack of smoothable curves. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: The white dots indicate that all singularities of given $(g,m)$ are smoothable stevens_la_rabida. The remaining dots indicate the existence of non-smoothable singularities of type $(g,m)$: black is due to Mumford mumford_pathologies, dark blue is due to Pinkham pinkham_thesis, light blue is due to Greuel greuel, and green is due to Stevens stevens_canonical_curves. The orange dots are novel to Theorem \ref{['thm:smoothability']}.

Theorems & Definitions (106)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 96 more